But Democratic Unionist Party (DUP) leader Arlene Foster poured cold water on the prospect, saying that the 1998 Good Friday Agreement sets out “criteria for a border poll, and it hasn’t been met – therefore it will not be called”.

What is the current Brexit state-of-play?

On Thursday, Mrs May met EU leaders in Brussels in a bid to secure changes to the Irish border backstop in the Brexit agreement.

DUP leader Arlene Foster said her party had a “wide-ranging” discussion with the taoiseach.

Mrs Foster also said some people were engaging in “project fear” with the Brexit negotiations.

The party’s deputy leader, Nigel Dodds, said the backstop “is the problem”, but would not specify which possible alternative his party is supporting.

Sinn Féin’s vice-president Michelle O’Neill said her party would “hold the taoiseach’s feet to the fire” when it comes to defending the backstop.

She said he had given her an assurance he would remain firm with his stance.

The party also said they have been calling repeatedly for a border poll, and that they had urged Mr Varadkar to begin planning for one.

The Ulster Unionist Party’s (UUP) Brexit spokesperson Steve Aiken said there needed to be “level-headed conversations” and that the UUP had told the taoiseach how concerned they are by the terms of the Irish border backstop.

The UUP said it is working on a number of alternative proposals it wants the UK and EU to consider.

Alliance Party leader Naomi Long said they had a very constructive and wide-ranging discussion with Mr Varadkar.

“It’s fairly clear those this week suggesting there is some chance of the UK and Irish government doing a side deal without the EU are chasing after a no-way scenario,” she said.

SDLP leader Colum Eastwood said that it had been a “good meeting” and added that he and the taoiseach are “on the same side of this argument”.

“We have been watching with some dismay what has been going on in Westminster over the last couple of months,” he said.

“I don’t think anybody within the Irish government or the European Commission sees any opportunity for diluting the protection of citizens in Northern Ireland.”

What happens next?

Brexit Secretary Steve Barclay and EU negotiator Michel Barnier will hold talks in Strasbourg on Monday, as the EU and UK Brexit negotiating teams discuss proposed changes to the deal.

British sources say the talks will include discussion of the legally-binding withdrawal agreement, the BBC’s Brussels reporter Adam Fleming said.

An EU source said the further talks are an opportunity to listen to the UK’s ideas.

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Your guide to Brexit jargon

Mrs May and Mr Juncker will meet again before the end of February, to review progress.

The prime minister is expected to put the deal to a vote in the Commons towards the end of February.

She said the plan must change if it is to win the support of MPs who urged her to seek “alternative arrangements” to the backstop when rejecting the deal last month.